= Half-final response of a graded incoherent feed-forward loop
{title2=$t_{1/2}$}
Let an output start at zero, degrade at rate $\alpha$, and switch from production $F\beta$ to $\beta$ at time $d$, with $F>1$. Its final level is $Z_s=\beta/\alpha$. Writing $\delta=1-e^{-\alpha d}$, its first crossing of $Z_s/2$ occurs at $t_{1/2}=\alpha^{-1}\ln[2F/(2F-1)]$ if $F\delta\ge1/2$; otherwise it occurs at $d+\alpha^{-1}\ln[2(1-F\delta)]$. This follows by solving the two linear <differential equations> before and after the switch. A large F alone does not make the first formula uniform in arbitrarily short pulse durations.
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